References & Citations

Bookmark

Mathematics > Algebraic Topology

Title:Fibred sites and stack cohomology

Abstract: The usual notion of a site fibred over a stack is expanded to a definition of
a site C/A fibred over a presheaf of categories A. Presheaves of simplicial
sets on the site fibred over a presheaf of categories A are contravariant
enriched diagrams defined on A, taking values in simplicial sets. The standard
model structure for presheaves of simplicial sets induces a coarse equivariant
structure for enriched contravariant A-diagrams. If the presheaf of categories
is a presheaf of groupoids G, then the associated homotopy theory is Quillen
equivalent to the homotopy theory of simplicial presheaves over BG, and so the
homotopy theory for the fibred site C/G is an invariant of the homotopy type of
G. Similar homotopy invariance results obtain for presheaves of spectra and
presheaves of symmetric spectra on C/G. In particular, stack cohomology can be
calculated on the fibred site for a representing presheaf of groupoids.